I wanna prove following equation $ \sum_{i=1}^n \prod_{k=1,k\neq i}^n \prod_{j=1,j\neq k}^{n+1}(x_j - x_k) = -\prod_{i=1}^n \prod_{j=1,j\neq i}^n (x_j - x_i) $
I have verified several numbers such as $n=2,3,4$, and try to prove it using mathematical induction, however I can't extend the 2 $\sum \sum$. Some guru told me to try resultant, but it seems to be sums of resultants, and can't be simply used.
Can anyone help me on this? thanks a lot.
$\sum_{i=1}^n \prod_{k=1,k\neq i}^n \prod_{j=1,j\neq k}^{n+1}(x_j - x_k) = -\prod_{i=1}^n \prod_{j=1,j\neq i}^n (x_j - x_i)$
Denoting $A_i=\prod_{j=1,j\neq i}^n (x_j - x_i)$ we have
$$\sum_{i=1}^n \prod_{k=1,k\neq i}^n (x_{n+1}-x_k)A_k = -\prod_{i=1}^n A_i$$
From here $$\sum_{i=1}^n \frac{\prod_{k=1,k\neq i}^n (x_{n+1}-x_k)}{A_i} = -1$$
The last identity is just the Lagrange interpolation polynomial (http://en.wikipedia.org/wiki/Lagrange_polynomial).
